The Use of Geometric Diversity for Spectral Dominance in Underground Imaging

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1 The Use of Geoeic ivesiy fo pecal oiace i Udegoud Iagig Loezo Lo Moe, Rashid Asai, ailo icolo epae of lecical ad Copue gieeig Uivesiy of Illiois a Chicago Chicago, IL 667 looe@ece.uic.edu Michael C. Wics esos iecoae Ai Foce Reseach Laboaoy Roe, NY Michael.wics@l.af.il Absac Udegoud iagig of dielecic ad coducive aoalies pefoed usig goud peeaig adas (GPRs equies expesive widebad syses o icease he esoluio. The adve of oogaphic piciples i uli-oosaic GPRs daaically ipoved he iagig capabiliies ad suggesed he possibiliy of educig he badwidh of he pobig wavefo. I his wo we popose o exed he oogaphic piciples o he case of below-goud disibued sesig, hus aig advaage of he geoeic divesiy. We show ha, by usig geoeic divesiy, he fequecy coe equied o iage below-goud ages is dasically educed o viually a sigle oochoaic sigal, hus achievig full specal doiace i he wavefo desig. I. INTROUCTION Pesely, a pevale appoach o deec, locae, ace ad iage udegoud objecs is by usig Goud Peeaig Rada []-[5]. eph of peeaio is augeed by loggig daa hough boeholes. Iage esoluio is aelioaed by usig oogaphic piciples applied o he eceived daa [6]- [8]. Howeve, all GPR syses eed o use sho pulses (i.e. high fequecy badwidh o icease he ifoaio coceig he ages via fequecy divesiy. The use of high badwidh leads o seveal issues. Fis, he sigal o oise aio (NR deceases wih a icease i he specal coe of he pobig wavefield. ecod, he elecoageic specu available fo iliay ad civil applicaios is coiuously beig eoded due o he eedous dead of wieless applicaios. Fuheoe, uieioal (e.g. boadcasig saios o ieioal (e.g. jaes a-ade iefeeces ca educe he available specu. If lage badwidh is equied, MI, MC ad ieodulaio effecs becoe difficul ass o be acled ad solved. Thid, a widebad syse ca be exeely buly, delicae ad expesive. This poble is acceuaed whe he syse is desiged o wo a lowe fequecies: i is ipacical o geeae well-desiged sho pulses i he HF fequecy age, whee aeas ae fudaeally elecically sall. Addiioally, he shoe he pulse is, he geae is he difficuly ad cos i popely saplig he eceived sigal. Fouh, he coduciviy ad dielecic peiiviy of he goud vaies wih he fequecy: if a widebad sigal is se io he goud, fequecy dispesio is liely o occu, hus boadeig he pulse suppo ad educig he achievable esoluio. Addiioal o-badwidh elaed facos lii he efficiecy of coo GPRs. Fo exaple, he esoluio i aziuh depeds o he beawidh: a HF fequecies i is ipacical o ceae pecil beas, heefoe aziuh esoluio us be educed by usig ohe echiques. Moeove, coo GPR suffe fo he blid egio effec i which he eceive is idle uil he asie coplees he asissio of he pulse. This poble ca be solved by ivoig suiable odulaio echiques, a he pealy of iceased coplexiy ad cos of he syse. Fially, whe a age is o paallel o he suface, ule s law suggess ha efleced eegy is o aily bac-popagaig o he GPR eceive, ad hus is deecio ay be copoised. We popose a ehodology [9]-[4], aed RF oogaphy, ha addesses hese ope issues, ad we deosae how RF oogaphy ay oupefo cue sae-of-he-a GPR echology. RF oogaphy equies a se of low-cos peeable aspodes abiaily deployed above he goud (see Fig.. Tasies sed a wavefo io he goud, scaees e-iadiae powe owad eceives, which log daa ad elay he eieved ifoaio o a base saio. The ovely of his appoach is he use of uliple asies (i.e. view divesiy ad uliple eceives (i.e. obsevaio divesiy, besides fequecy, polaizaio ad aea pae divesiy. The accual of geoeic divesiy faciliaes he wavefo fequecy coe educio, ad i he liiig case a se of discee oochoaic sigals yield acioable below-goud 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

2 ecosuced iages. I his wo we focus o he ivesio algoihs ecessay o pocess he sigals easued by he uliude of goud sesos. = ω μ εε + jμ σ / ω = ω μ ε. (3 The fucio i ( accous fo he diffeece bewee he uow dielecic peiiviy of he objec ad ha of he hos ediu. Fo each poi holds: ' i egio, he veco wave equaio ( ' ε ( ' ( ' = +. (4 Figue : RF Toogaphy. Tasies sed powe io he goud. Receives collec he scaeed field ad sed his ifoaio o he ai saio. II. FORWAR MOL We descibe he fowad odel of a RF oogaphic syse by cosideig he 3 geoey depiced i Fig.. The hos ediu (i.e. he eah is odeled as a hoogeeous ediu wih elaive dielecic peiiviy ε, coduciviy σ, ad ageic peeabiliy μ. The ages ae assued o eside i he ivesigaio doai. The souces ae N elecically sall dipoles (of legh Δ l o loops (of aea A fed wih cue I, ad locaed a posiio (view divesiy. Fo each asiig aea, he scaeed field is colleced by M eceives (obsevaio divesiy, locaed a pois i space. Fo sipliciy, a sigle opeaig fequecy f is adoped. We assue he elaive dielecic peiiviy pofile ε ( ' ad he coduciviy pofile ( ' σ iside he ivesigaio doai as uows of he poble. Accodigly, he ivese poble is ecas i es of he uow peiiviy coas fucio: ( ' σ σ ε ( ' = ε( ' ε + j. ( π f ε I his way, he wave ube iside ca be expessed as: ( ' = ωμεε ( ' + jωμσ ( ' = + ε ( ', ( The scaeed wave i a poi ha is soluio of (4 ca be wie i es of iegal equaio of he dyadic Gee s fucio: ( = ( ( ( G, ' ' ' d ', (5 whee ( ' is he oal field i he ivesigaio doai, I give as he supeposiio of he icide field ( ' (i.e. he field i he ivesigaed aea whe objecs ae abse ad he field (, scaeed by he ages. As i is well ow, he ivese scaeig poble i (5 is o-liea. Neveheless, i ca be ecas o a liea poble by eas of he Bo appoxiaio (BA. Ude BA, he oal field iside he iegad of (5 ca be appoxiaed by he ow icide field [5]-[8], i.e.: ε I ( ( ( ( G, ' ' ' d '. (6 Theefoe, he ivese poble a had is cas as he ivesio of he liea iegal equaio coecig he peiiviy coas fucio o he scaeed field daa. The use of BA ca be jusified by cosideig ha: The ages of iees ae isolaed, liied i ube ad ebedded i a lossy ediu. Theefoe, uual ieacio (a pheoeo igoed by BA bewee ages ca be assued egligible. I geeal, he ihoogeeiies of he soil ae elecically sall, ad hei coduciviy eais low. Theefoe, hei scaeed fields ae isigifica copaed wih he RF sigal e-iadiaed by ou ages of iees. Ou goal is o deec, localize ad appoxiaely deeie he geoey of he ages. Towad his objecive i has bee show how BA based ivesio algoihs ae able o wo wih sog scaeig objecs, povided ha o quaiaive descipio of he dielecic peiiviy i is equied. ε 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

3 The Gee s fucio depeds upo he geoey ha is cosideed (half-space, wo-diesioal, full space. I his wo, we uilize he hee diesioal dyadic Gee s fucio G fo a hoogeeous ediu wih he sae popeies of he eah. This assupio is easoable because he sesos ae deployed a he ai/goud ieface, ad he fequecies ivolved ae elaively low. Accodigly, we have [9]: (, ' G j ' e = I + 4 π '. (7 The opeao i (7, which is esposible fo depolaizaio ad is useful fo ea field sesig, ca be geeally egleced fo ' >>, i.e. whe sesos ae locaed i he fa zoe wih espec o ages. The icide field, i.e. he field adiaed i he hoogeeous ediu fo a poi souce locaed a posiio, is give by: ( ', = Q ( ', ˆ G a, (8 I = ωμ Δ fo a elecically sall dipole, o whee Q j l I Q = jωμa I fo a elecically sall loop, a ˆ is he (elecic o ageic dipole oe diecio. Addiioally, he field eceived by a dipole o loop wih oe diecio ˆ a posiioed a due o a equivale (i es of I cue disibuio defied iside he ivesigaio doai ca be expessed as [8]: (, = I ( ( ε ( aˆ G, ' ', ' d '. (9 ubsiuig (8 i (9 we obai he scala fowad odel of RF oogaphy: (, = L( ε ( ' = Q ( ( ε ( a G, ' G ', a ' d '. ( Fo a aheaical poi of view, he poble of fidig he coas fucio is o pefo he ivese of he liea opeao L coecig he uow coas fucio ad he scaeed field daa. Figue : 3 Geoey fo he ivesio odel. III. INVRION PROCUR A. Tihoov Regulaizaio A way o copue L is o pefo a ueical ivesio of L []. Le us collec he sapled field daa i a odeed { } NM veco (, =, ad disceize he doai egio i K voxels, each oe locaed a posiio : he coas dielecic peiiviy ca be ebodied i a colu { } veco ε ε ( ' = of legh K, ad i epeses he se of uow paaees. Afe his disceizaio, eq. ( ca be ewie i a aix fo: = L ε, ( whee L ow is a aix wih diesios NM K. The poble is he o ive he elaio (. ue o he idepede se of easuees, L is heoeically full a, bu is ofe seveely ill-codiioed. This leads o sevee aifacs i he ecosucio pocess, paiculaly exacebaed whe oise (heal, exeal, quaizaio o clue is ipigig he eceives. A coo way o quaify he behavio of L is by ispecio of is codiio ube κ. Fo he opeao L i 6 is quie coo o obai ypical values of κ above. A efficie ehod o pefo a ivese of a vey ill codiioed aix is by usig he Tihoov egulaizaio pocedue. I his way, he coas dielecic peiiviy ca be esiaed: ( H H ˆ = + ε LL βi L ( ' 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

4 H Whee L deoes he adjoi of L, ad β is he egulaizaio paaee i he Tihoov sese, ha eeds o be appopiaely seleced. The advaage of his appoach is is eaable pefoace i geeaig eaigful iages, eve whe he ube of sesos is liied. Ufouaely, a pope choice of β ay be a difficul as, ad ofe i is ecessay o see fo a cosaied opiizaio soluio of β befoe a eaigful, shap ad low blued iage is ecosuced. This iplies a (copuaioally expesive aix ivesio fo each aep ay be ecessay. B. Fouie Appoach I pacical sceaios, whee eal-ie pocessig is ciical, o whe he elay o a base-saio is ipeded, i is pivoal o deive a ivesio saegy ha pivileges speed vs. accuacy. This pioiy is ephasized by he cue syse echology, which is widely ipleeig FFT ouies o acceleae iage ecosucios. Theefoe, we popose a appoach ha aes advaage of he Fouie elaio aisig bewee scaeed field ad objec shape, as discussed i lieaue ude he opic of diffacio oogaphy. I fac, if ages ad sesos ae disa eough so ha he popagaig wave is TM (oally occuig whe he fields ae piaily popagaig as /, he he fowad odel ca be expessed as follows below. We defie he ui o diecio of popagaio vecos as: ˆ l ˆsi cos ˆsi si ˆ = x θ ϕ y θ ϕ z cosθ, (3 ˆ l = xˆsiθ cosϕ + yˆsiθ siϕ + z ˆcosθ. (4 Usig he paaxial appoxiaio, he asiig Gee s fucio a he geeic posiio ' iside egio ca be siplified as: G (, ' ( ( ˆ + j + jl exp exp ', (5 4π while he eceivig Gee s fucio ca be expessed as: G ( ', ( ( ˆ + j jl exp exp '. (6 4π Theefoe, fo a pai of asies ad eceives, he scaees field ca be ewie as: (, 6π ( aˆ aˆ + j + Qe ε ( ( ˆ ˆ + j 'exp l l ' d' q. (7 is a siplified ad coiuous vesio of (. The quaiy ( ˆ ˆ veco:.(7 l l ca be epeseed by a 3 ( ˆ ˆ = l l. (8 q. (7 ca be ewie as [33]: ( 6π ( aˆ aˆ + j + = Qe ε ( ( + j 'exp ' d' I is useful o coside a oalized vesio of (9: ˆ ˆ a a ( 6π j + ( = e Q ( ( j. (9 ( = ε 'exp + ' d' ( This esul ca be iepeed i he followig way: each colleced saple ( eus he value of he specal copoe of he coas fucio ( ' ε. Theoeically, if we have eough saples o fully populae he specal epeseaio of ( ' ( fucio ( ε, he discee fucio i he lii ca be appoxiaed as a coiuous, ad ( ca be iepeed as a 3 ivese Fouie asfo of he peiiviy coas fucio. Theefoe, we ca ecosuc a iage of he udegoud by diec Fouie asfo eq. (9, i.e.: ( ( ( ˆ ε ' exp j ' d, ( = K whee he doai of iegaio K is he suppo of. By ispecio of (8, we coclude ha whe he ( sesos copleely ecicle he age, K is a sphee of adius, eaig ha he available ifoaio of he 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

5 specal coe of ( ' ε is liied up o he specal copoe. Theefoe, he ecosuced iage of he coas fucio will be a low-pass fileed vesio of he ue iage. I he eal sceaio whee a fiie ube of sesos ae deployed, hee facos affec he esoluio (leadig o bluig ad aifacs: he ivalidiy of paaxial appoxiaio, he o-uifo saplig, he spase daa se, ad he aeuaio cosa. I his wo, we coside he aeuaio o be egligible, so ha he wave ube i ( eais a eal quaiy, ad FFT ca be applied. Paaxial appoxiaio holds whe he agle θ bewee he ay passig hough he oigi ad he ay ax iesecig he bouday of he egio is egligible. This agle ca be copued usig: θ ( ' ˆ ˆ j ( ' ' l l j ax = ax ax a j ' ˆ ˆ ˆ j l j + l j l j, ( whee j epeses ay asie o eceive. Blu educio is accoplished by segeig he egio io salle aalysis egios (whee θ ax eais sall wihi he subegio ad by cosideig a ivese poble (i.e. salle FFT fo each sub-egio. The, he esulig sub-iages ae cocaeaed o fo he fial iage. The o-uifo o uifo gid asfoaio ca be accoplished usig Ti-Liea iepolaio. Le us defie a uifo gid i he specal doai of he scaeed field (,,, whee: Nxyz,, Nxyz,,,, = Δxyz,,, + Δxyz,,,... Nxyz,,, Δxyz,, Fo he i-liea iepolaio, le us defie hee ievals i (3 he Fouie space: Δ, Δ, Δ. Le us coside a saple poi (,, i he uifo gid o be esiaed, ad a o-uifo saple poi = + + ha has bee easued: we ca defie a iepolaio weighig faco as follows: ( ( ( w = h u h v h w (4 whee: x y z h xyz,, ( a The oal weighig faco is: a a Δ = Δ W x, yz, xyz,, (5 ohewise N M = w (6 = = The esiaed value i (,, is heefoe: N M u v w w (,, ( = (7 W = = The ajo advaage of his echique is he iisic possibiliy of esiae issig saples whe we choose Δ x, yz, >Δxyz,,. I his way, he ecosuced iage shows fewe aifacs ad fewe oscillaios. A way o ecove ifoaio fo he issig saples o he spase daase ca be accoplished by usig he echique of Pojecio o Covex se (POC. The basic idea of POC is o popely weigh he available saples i a way ha he coespode poi spead fucio is iiized. The ideal poi spead fucio (PF ( ' ( jux' + vy ' + wz ' = Δ Δ Δ (8 PF e u + v + w coicides wih he ipulse espose of he RF oogaphy opical syse, i.e. has he shape of a 3 Bessel-sic fucio, ad i ca be copued ueically by usig a 3 FFT algoih so ha he PF is ow fo ay value of ' i he egio. Fo sipliciy, le us coside a equivale scaled poble i which,, ad Δ Δ Δ = Howeve, he acual PF geeaed by he available saples is: N N N ( ' = s s s s s= jux ( ' vy' wz' (9 PF p e + + whee he weighig facos p s ae ideically equal o uiy. I piciple, by popely weighig he elees i he icoplee suaio (9, i ay be possible ha he weighed acual PF is poiwise vey siila o he ideal PF: 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

6 ( ', ', ' PF ( x ', y ', z ' PF (3 I ode o achieve his esul we ivoe he Pojecio o Covex es ehod, which is based o successive appoxiaios of he acual PF uil a soppig cieio is e. I he fis ieaio, we esiae he poi spead fucio by iposig all weighs o be p =. The ex sep is o copae poiwise he copued PF wih he ideal PF (i he pace doai. Moe exacly, we ceae a suiable guad bad egio o he ideal PF. If he values of he acual PF ae fallig ouside he guad egio, we foce hese values o be iside he guad egio. Oce he coecios o he copued PF ae doe, we pefo a 3 IFFT. The Fouie asfo of he coeced acual PF is geeally a fucio ha has values fo ay,, pais. The ex sep is o se o zeo all pois i he Fouie doai ha epese ou issig saples. I his way, we ae ceaig a ew fucio defied oly whee acual saples ae locaed. We pefo a ew copuaio of PF usig he values peviously obaied, ad he POC ieaio coiues fo uil a soppig cieio is e, e.g. he copued PF lies copleely wihi he guad bad egio, o he -h ieaio of he copued PF does o ipove he appoxiaio of he ideal PF wih espec of he -h ieaio, i.e. he pocess salls. Whe POC eiaes, he coefficie p s of he las ieaio ae used fo copuig he o-uifo ivese Fouie asfo of he eceived elecic field, as follows: p s Figue 4: ecosuced iage usig Fouie, o POC Figue 5: ecosuced iage usig Fouie ad POC jux ( s ' + vy s ' + wz s ' ˆ ε ( ' = pse ( us, vs, ws (3 s= IV. IMULATION AN RULT We pefoed seveal siulaios usig he ehods descibed above. The geoey is depiced i Fig. 3, he pobig fequecy is 3MHz, ad esuls ae show i Figs. 4,5,6. Cosideaios ad fuhe exaples will be show a he ie of he cofeece. Figue 6: ecosuced iage usig Tihoov Figue 3: : Geoey fo he siulaio (op view. Tasies ae epeseed wih +, while eceives ae epeseed wih X. The wo blac lies epese he posiios of he uels. V. ACKNOWLGMNT The auhos ae haful o M. W. Baldygo, Ai Foce Reseach Laboaoy, fo sposoig ad fudig his eseach. We ae also gaeful o M. J. Pae,. M. Feaa, Ai Foce Reseach Laboaoy,. F. oldoviei, Cosiglio Nazioale Riceche, Ialy, Pof. M. Cheey, Resselae Polyechic Isiue, fo hei echical discussios. 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.

7 RFRNC [] R.J. Lyle,.F Laie,.L. Lage ad.j. avis Usig Coss Boehole lecoageic Pobig o Locae High Coas Aoalies, Geophysics. Vol. 44, pp , 979. [] A.J evaey, Geophysical iffacio Toogaphy I Tas o Geosci. Reoe es. Vol. G-, No., pp. 3-3, Ja [3]. J. aiels, Goud Peeaig Radas, ecod diio. Isiuio of lecical giees, Lodo, UK, 4. [4] A.J. Wie, J.. Molyeux, J.. Nyquis, Goud Peeaig Rada Toogaphy: Algoihs ad Case udies, I Tas. Geosci. Reoe es., Vol 3, No., pp , Ma [5] A.J. Wie, ad. Log, hallow Applicaios of Geophysical iffacio Toogaphy. I Tasacios o Geosciece ad Reoe es., Vol. 4, No. 5, p , ep. 986 [6] T. B. Hase, ad P. M. Johase, Ivesio chee fo Goud Peeaig Rada ha Taes Io Accou he Plaa Ai-oil Ieface, I Tas. Geosci. Reoe es., Vol. 38, No., pp , Ja. [7] T. J. Cui ad W. C. Chew, iffacio Toogaphic Algoih fo he eecio of Thee-iesioal Objecs Buied i a Lossy Half- pace, I Tasacios o Aeas ad Popagaio. Vol. 5, No., pp. 4-49, Ja.. [8] P. Meice, Liea GPR Ivesio fo Lossy oil ad a Plaa Ai- oil Ieface, I Tas. Geosci. ad Reoe es., Vol 39, No., pp. 73-7, ec. [9] M. C. Wics, RF Toogaphy wih Applicaio o Goud Peeaig Radas, I Poc. 4 s Asiloa Cofeece ACC 7, pp. 7-, 4-7 Nov. 7 [] L. Lo Moe,. icolo, ad M. C. Wics, Popagaio Model, Opial Geoey ad Receive esig fo RF Geooogaphy, I Poc. RadaCo 8, Roe, Ialy, May 6-3, 8. [] L. Lo Moe, A. M. Bagci,. icolo, ad R. Asai, paial Resoluio i Toogaphic Iagig wih iffaced Fields, Poc. XXIX Geeal Assebly of he Ieaioal Uio of Radio ciece (URI, Chicago, IL, UA, Aug. 7-6, 8. [] L. Lo Moe, ad. icolo, isibued RF Toogaphy fo Voids eecio, Poc. 8 Meeig fo he Miliay esig yposia (M, pecialy o Balespace Acousic ad eisic esig, Mageic ad lecic Field esos, Lauel, M, UA, Aug. 9-, 8. [3] J. Nogad, M.C. Wics, ad A. ozd, isibued/bedded ub- uface esos fo Iagig Buied Objecs wih Reduced Muual Couplig ad uppessed lecoageic issios, Poc. Ieaioal Cofeece o lecoageics i Advaced Applicaios (ICAA, pp , Tui, ep [4] L. Lo Moe, ad. icolo, Receivig Aea esig fo Goud Peeaig Toogaphic Iagig, Poc. I AP- Ieaioal yposiu, a iego, CA, UA, Jul. 5-, pp. -4, 8. [5] G. Leoe, F. oldoviei, Aalysis of he disoed Bo appoxiaio fo ubsuface Recosucio: Tucaio ad Uceaiiies ffec, I Tas. o Geosci. ad Reoe es., Vol. 4, No., pp , Ja. 3 [6] R. Pesico, R. Beii, F. oldoviei, The Role of he Measuee Cofiguaio i Ivese caeig fo Buied Objecs ude he Bo Appoxiaio, I Tas o Aeas Popag., Vol. 53, No. 6, pp , Ju 5 [7] H. J. Li ad F. L. Li, A Geealized Iepeaio ad Pedicio i Micowave Iagig Ivolvig Fequecy ad Agula ivesiy, J. lecoageic Waves ad Applicaios, Vol. 4, No. 5, pp , 99 [8] F. oldoviei, J. Hugeschid, R. Pesico, G. Leoe, A Liea Ivese caeig Algoih fo Realisic GPR Applicaios, Nea uface Geophysics, Vol. 5, No., pp. 9-4, Feb. 7 [9] W.C. Chew, Waves ad Fields i Ihoogeeous Media. I Pess, 995, Piscaaway NJ [] M. Beeo, ad P. Boccacci, Ioducio o Ivese Pobles i Iagig, Isiue of Physics Ld, Lodo, UK,. 9 Ieaioal W& Cofeece /9/$5. 9 I Auhoized licesed use liied o: WAHINGTON UNIVRITY LIBRARI. owloaded o Ocobe 6, 9 a 5:6 fo I Xploe. Resicios apply.